Biomedical Engineering Reference
In-Depth Information
where the difference in the atomic binding of the electrons has been neglected
in writing the last equality. In terms of the values
P
and
D
of the parent and
daughter, the energy released in positron decay is given by
=
P
-
D
-2
mc
2
.
Q
β
+
(3.41)
Therefore, for positron emission to be possible, the mass of the parent atom must
be greater than that of the daughter by at least
2
mc
2
1.022
MeV. Using the infor-
mation from Appendix D, we find for the energy released via positron emission in
the decay (3.37) to the ground state of
2
10
Ne
Q
β
+
=
=
-5.182 - (-8.025) - 1.022
=
1.821 MeV.
(3.42)
Electron capture, which results in the same net change as positron decay, can com-
pete with (3.37):
0
-1
e+
2
11
Na
22
10
Ne +
0
ν
(3.43)
→
.
Neglecting the electron binding energy in the
2
11
Na atom, we obtain from Eq. (3.35)
for the energy released by electron capture
Q
EC
=
-5.182 + 8.025
=
2.843 MeV.
(3.44)
[Comparison of Eqs. (3.35) and (3.41) shows that the
Q
value for EC is greater than
that for
β
+
decay by 1.022 MeV when
E
B
is neglected.]
We next develop the decay scheme for
2
11
Na. Appendix D indicates that
β
+
emis-
sion occurs 89.8% of the time and EC 10.2%. A gamma ray with energy 1.275 MeV
occurs with 100% frequency, indicating that either
β
+
emission or EC leaves the
daughter nucleus in an excited state with this energy. The positron decay scheme
is shown in Fig. 3.12(a) and that for electron capture in (b). The two are combined
in (c) to show the complete decay scheme for
2
11
Na. The energy levels are drawn
relative to the ground state of
2
10
Ne as having zero energy. The starting EC level is
2
mc
2
higher than the starting level for
β
+
decay.
Additional radiations are given in Appendix D for
2
11
Na. Gamma rays of energy
0.511 MeV are shown with 180% frequency. These are annihilation photons that
are present with all positron emitters. A positron slows down in matter and then
annihilates with an atomic electron, giving rise to two photons, each having energy
mc
2
0.511
MeV and traveling in opposite directions. Since a positron is emitted
in about 90% of the decay processes, the frequency of an annihilation photon is
1.8 per disintegration of a
2
11
Na atom. The remaining radiation shown, Ne X rays,
comes as the result of the atomic-shell vacancy following electron capture.
As this example shows, electron capture and positron decay are competitive
processes. However, whereas positron emission cannot take place when the
parent-daughter atomic mass difference is less than 2
mc
2
, electron capture can,
the only restriction being
P
-
=
D
>
E
B
, as implied by Eq. (3.35). The nuclide
12
53
I
can decay by three routes: EC (60.2%),
β
-
(36.5%), or
β
+
(3.3%).
13)
13 In general, the various possible decay modes
for a nuclide are those for which
Q
>0fora
transition to the daughter ground state.
